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4.3 Explicit form of the Einstein–Langevin equation

We can write the Einstein–Langevin equation in a more explicit form by working out the expansion of ˆab ⟨T [g + h]⟩ up to linear order in the perturbation hab. From Equation (19View Equation), we see that this expansion can be easily obtained from Equation (23View Equation). The result is
∫ ∘ ------ ⟨ˆTab[g + h; x)⟩ = ⟨ˆTab[g,x)⟩ + ⟨Tˆ(1)ab[g,h; x)⟩ − 1 dny − g(y)Habcd[g;x,y)h (y) + 𝒪 (h2).(31 ) n n n 2 n cd
Here we use a subscript n on a given tensor to indicate that we are explicitly working in n dimensions, as we use dimensional regularization, and we also use the superindex (1) to generally indicate that the tensor is the first order correction, linear in hab, in a perturbative expansion around the background gab.

Using the Klein–Gordon equation (2View Equation), and expressions (3View Equation) for the stress-energy tensor and the corresponding operator, we can write

( ) ˆT(1)ab[g,h ] = 1-gabhcd − δa hb− δbha ˆTcd[g] + ℱ ab[g,h ] ˆφ2[g], (32 ) n 2 c d c d n n
where ab ℱ [g;h] is the differential operator
( ) ( ) ℱ ab ≡ ξ − 1- hab − 1gabhc □ 4 2 c ξ [ + -- ∇c∇ahbc + ∇c ∇bhac − □hab − ∇a ∇bhcc − gab∇c ∇dhcd + gab□hcc 2 ( a b b a ab ab d ab d) c ab c d] + ∇ hc + ∇ hc − ∇ch − 2g ∇ hcd + g ∇ch d ∇ − g hcd∇ ∇ . (33 )
It is understood that indices are raised with the background inverse metric gab, and that all the covariant derivatives are associated to the metric gab.

Substituting Equation (31View Equation) into the n-dimensional version of the Einstein–Langevin Equation (14View Equation), taking into account that gab satisfies the semiclassical Einstein equation (7View Equation), and substituting expression (32View Equation), we can write the Einstein–Langevin equation in dimensional regularization as

[ ( ) ] --1--- (1)ab 1- ab cd ac b bc a ab 1-ab c 8πGB G − 2 g G hcd + G hc + G hc + ΛB h − 2g hc ( ) − 4αB- D (1)ab − 1gabDcdhcd + Dachb + Dbcha 3 2 c c ( 1 ) − 2βB B(1)ab − -gabBcdhcd + Bachbc + Bbchac 2 ∫ −(n−4) ab ˆ2 1- n ∘ ------ −(n−4) abcd − μ ℱx ⟨φ n[g;x )⟩ + 2 d y − g(y )μ H n [g;x,y)hcd(y) − (n− 4) ab = μ ξn , (34 )
where the tensors Gab, Dab, and Bab are computed from the semiclassical metric gab, and where we have omitted the functional dependence on gab and hab in (1)ab G, (1)ab D, (1)ab B, and ℱ ab to simplify the notation. The parameter μ is a mass scale which relates the dimensions of the physical field φ with the dimensions of the corresponding field in n dimensions, (n−4)∕2 φn = μ φ. Notice that, in Equation (34View Equation), all the ultraviolet divergences in the limit n → 4, which must be removed by renormalization of the coupling constants, are in ⟨ˆφ2n(x)⟩ and the symmetric part HabScnd(x,y ) of the kernel Hanbcd(x,y), whereas the kernels N nabcd(x,y) and Habcd(x, y) An are free of ultraviolet divergences. If we introduce the bi-tensor F abcd[g;x,y) n defined by
⟨ ⟩ F anbcd[g;x,y) ≡ ˆtanb[g;x )ˆtρnσ[g;y ) , (35 )
where ˆab t is defined by Equation (12View Equation), then the kernels N and HA can be written as
N abcd[g;x, y) = Re Fabcd[g;x, y), Habcd[g;x, y) = Im Fabcd[g;x, y), (36 ) n n An n
where we have used that
ab cd ab cd ab cd 2⟨ˆt (x )ˆt (y)⟩ = ⟨{ˆt (x),ˆt (y)}⟩ + ⟨[ˆt (x),ˆt (y)]⟩,

and the fact that the first term on the right-hand side of this identity is real, whereas the second one is pure imaginary. Once we perform the renormalization procedure in Equation (34View Equation), setting n = 4 will yield the physical Einstein–Langevin equation. Due to the presence of the kernel abcd H n (x, y), this equation will be usually non-local in the metric perturbation. In Section 6 we will carry out an explicit evaluation of the physical Einstein–Langevin equation which will illustrate the procedure.

4.3.1 The kernels for the vacuum state

When the expectation values in the Einstein–Langevin equation are taken in a vacuum state |0⟩, such as, for instance, an “in” vacuum, we can be more explicit, since we can write the expectation values in terms of the Wightman and Feynman functions, defined as

( ) G+n [g;x, y) ≡ ⟨0|φˆn [g; x)ˆφn[g;y)|0⟩, iGFn [g; x,y) ≡ ⟨0|T ˆφn[g;x)φˆn [g;y ) |0⟩. (37 )
These expressions for the kernels in the Einstein–Langevin equation will be very useful for explicit calculations. To simplify the notation, we omit the functional dependence on the semiclassical metric gab, which will be understood in all the expressions below.

From Equations (36View Equation), we see that the kernels abcd Nn (x,y) and abcd H An (x, y) are the real and imaginary parts, respectively, of the bi-tensor Fnabcd(x,y). From the expression (4View Equation) we see that the stress-energy operator Tˆab n can be written as a sum of terms of the form { } 𝒜 ˆφ (x ), ℬ φˆ (x) x n x n, where 𝒜 x and ℬ x are some differential operators. It then follows that we can express the bi-tensor F anbcd(x,y) in terms of the Wightman function as

F abcd(x,y) = ∇a ∇c G+ (x,y )∇b ∇d G+ (x,y ) + ∇a ∇d G+ (x,y)∇b ∇c G+ (x,y ) n x y (n x y n ) x y n( x y n ) +2 𝒟abx ∇cyG+n(x,y)∇dyG+n(x,y ) + 2𝒟cdy ∇axG+n(x,y )∇bxG+n(x, y) ab cd( +2 ) +2 𝒟 x 𝒟y Gn (x, y) , (38 )
where 𝒟ab x is the differential operator (5View Equation). From this expression and the relations (36View Equation), we get expressions for the kernels Nn and HAn in terms of the Wightman function + G n (x, y).

Similarly the kernel abcd HSn (x,y) can be written in terms of the Feynman function as

[ HaSbncd(x,y) = − Im ∇ax∇cyGFn (x,y)∇bx∇dyGFn (x,y) + ∇ax∇dyGFn(x, y)∇bx∇cyGFn (x,y ) ab e c x d − g (x)∇ x∇ yGFn(x, y)∇e∇ yGFn (x,y ) − gcd(y)∇a ∇eG (x,y )∇b ∇y G (x,y) x y Fn x e Fn + 1gab(x)gcd(y)∇e ∇f G (x,y)∇x ∇y G (x,y) 2 x y Fn e f Fn + 𝒦ab (2∇c G (x, y)∇d G (x, y) − gcd(y)∇e G (x,y)∇y G (x,y)) x ( y Fn y Fn y Fn e Fn ) + 𝒦cdy 2∇axGFn (x,y)∇bxGFn (x, y) − gab(x )∇exGFn (x,y)∇xeGFn (x,y) ab cd( 2 )] +2 𝒦 x 𝒦 y G Fn(x,y) , (39 )
where ab 𝒦x is the differential operator
ab ( ab a b ab ) 1- 2 ab 𝒦 x ≡ ξ g (x)□x − ∇ x∇ x + G (x) − 2 m g (x ). (40 )
Note that, in the vacuum state |0⟩, the term ˆ2 ⟨φn(x)⟩ in Equation (34View Equation) can also be written as ⟨ˆφ2n (x )⟩ = iGFn (x,x) = G+n(x, x).

Finally, the causality of the Einstein–Langevin equation (34View Equation) can be explicitly seen as follows. The non-local terms in that equation are due to the kernel H (x, y) which is defined in Equation (22View Equation) as the sum of H (x,y ) S and H (x,y) A. Now, when the points x and y are spacelike separated, ˆφ (x ) n and ˆ φn (y) commute and, thus, + 1 ˆ ˆ Gn (x,y) = iGFn(x, y) = 2⟨0|{φn(x),φn (y )} |0⟩, which is real. Hence, from the above expressions, we have that HabAcnd(x,y ) = HabScnd (x,y) = 0, and thus Habncd(x,y) = 0. This fact is expected since, from the causality of the expectation value of the stress-energy operator [282Jump To The Next Citation Point], we know that the non-local dependence on the metric perturbation in the Einstein–Langevin equation, see Equation (14View Equation), must be causal. See [169Jump To The Next Citation Point] for an alternative proof of the causal nature of the Einstein–Langevin equation.


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